Question

Which of the following flows are represented by the velocity field, \[ \mathbf{V} = (x^2 - y^2) \hat{i} - 2xy \hat{j}? \]

• A. Incompressible flow
• B. Compressible flow
• C. Irrotational flow
• D. Rotational flow

Hint:

To identify if a flow is rotational or irrotational, check the curl of the velocity field. A zero curl indicates irrotational flow.

Answer 1

The Correct Option is A, C

Step 1: Understanding the flow characteristics.
The given velocity field is \(\mathbf{V} = (x^2 - y^2) \hat{i} - 2xy \hat{j}\). To determine whether the flow is incompressible, compressible, irrotational, or rotational, we need to check for the divergence and curl of the velocity field.
Step 2: Check the divergence.
The divergence of a flow is given by: \[ \nabla \cdot \mathbf{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} \] For the given velocity field, \[ \frac{\partial}{\partial x}(x^2 - y^2) = 2x, \quad \frac{\partial}{\partial y}(-2xy) = -2x \] Thus, \[ \nabla \cdot \mathbf{V} = 2x - 2x = 0 \] Since the divergence is zero, the flow is incompressible.
Step 3: Check the curl.
The curl of the flow is given by: \[ \nabla \times \mathbf{V} = \left(\frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\right) \hat{k} \] For the given velocity field: \[ \frac{\partial}{\partial x}(-2xy) = -2y, \quad \frac{\partial}{\partial y}(x^2 - y^2) = -2y \] Thus, \[ \nabla \times \mathbf{V} = (-2y + 2y) \hat{k} = 0 \] Since the curl is zero, the flow is irrotational.
Step 4: Conclusion.
Therefore, the correct answer is (D) Rotational flow.